I have a problem where I forget the proof of a theorem after some time without reworking it out. However, my teacher said that he was able to prove a theorem even without reworking it out for a long time. This puzzles me: is it I am too incompetent, or my teacher is too smart ?
[Math] How to remember all the proofs in mathematics
proof-explanationproof-writingsoft-question
Related Solutions
One important thing about proofs is that you will never be able to appreciate them, and therefore to learn from them, if you are not capable of reading the statement to be proved with a sceptical attitude, and to try to imagine it is untrue:
What's this nonsense they are claiming, it cannot be true!
Certainly it must be possible to satisfy the hypotheses without being obliged to accept the conclusion!
Once you have some mental idea of what a counterexample to the statement would look like, you can interpret the proof as an argument that systematically talks this idea out of your head, convincing you that it really is not possible to ever come up with such a counterexample.
Then you will have acquired a feeling of what the proof is really about, and you will be far more likely to retain it, and to come up with similar arguments when you need to prove something yourself.
But if you take a docile attitude and accept the statement to be proved from the onset, you will never be able to understand what all this reasoning was needed for in the first place.
It's perfectly normal. In fact, I think that's how a mathematitian's mind grows.
- First, you are naive and "intuistic", and you do a lot of "well, of course this is so!" like statements that are not well founded.
- After you are repeatedly hit over the head with examples where your intuition fails, you take a huge step back. You realize that even simple statements may be wrong, and that you need a rigorous way of proving them. You use strict logic notation to avoid any and all confusion.
- When you are more and more practiced, you begin to transition back a little. Yes, you still know that every statement has a strict logical form, but you don't write it down anymore. You begin, again, to rely on intuition.
At least, that's how it's worked for me. Mind you, the intuition in the final step is very different than the original intuition. The original one is the brash "d'ah, how can you ever doubt that?!" kind of thing that is embarasingly bad at doing math.
The more developed intuition in step 3 is much different. It has a lot more thought put into it. It's more "yeah, this is so, and I know approximately how I can prove this using strict logic, but since it would take 3 pages, I won't use strict logic."
Sure, the second intuition can also be wrong. And every once in a while, it is. But its performance is way way way way way better than in the beginning, and it also has a fall-back. If all else fails, your final answer is no longer "well, but, but how could it not be true?", the final answer is "oh alright fine, I'll spell it out rigorously!"
Best Answer
Your teacher is probably pretty smart, but that's not why he can prove the theorem and you can't. The difference between you two lies somewhere else:
You only saw the proof once, and you also probably only saw somewhere between several ten and a hundred proofs in your life. Your professor proved the theorem in question once every year for several years, and also probably encountered several thousand other proofs in his life.